A Blog on Science, Medicine, and Education

Packing Puzzle

This month, guest blogger Ari Boltax shares her thoughts about various kinds of packing and folding problems. Ari is a recent graduate of Brandeis University and is currently a student at Cornell Veterinary School.

After our winter break, my housemate brought home a packing puzzle. It consists of eighteen blocks in three different shapes. The challenge is to fit all eighteen pieces into a cube. Such a seemingly simple task like “make a cube out of this” kept me at the kitchen table at least an hour every day for five days piling blocks into the box, only to be sorely disappointed when I couldn’t make order from the disorder. One night I was trying the puzzle with a friend, and she joked that maybe if we randomly play around with it enough, we’ll just happen upon the solution. “Sounds like a plan,” I joked, “proteins do it all the time when they fold up, so why can’t we?”

It’s certainly true that the majority of what happens inside of cells is due to random collisions. Most of the time when two molecules bump into each other, nothing remarkable comes of it. That’s because the two molecules have to be in exactly the right orientation with exactly the right amount of energy. For proteins, it seems vastly unlikely that the right amino acids in a chain will ever attract for long enough to make a fully functional protein because there are just so many options. There can be as many as 34,350 amino acids in a single protein, as is the case for the molecular spring, Titin, which is found in every muscle that you move voluntarily.

The enormity of the protein packing puzzle was quantified in 1969 by a molecular biologist, Cyrus Levinthal, who estimated that there are over 10143 possible ways that a protein could fold. Sure, molecules move a lot faster than we do, but if proteins actually sampled every possible option before folding into their final conformations, they would take way more time than it actually takes for a protein to fold (a couple of microseconds, on a good day). There has to be something else – an underlying set of rules – because by Levinthal’s metric, it’s a wonder that any protein folds at all, and I shudder to think of the chaos that goes on in a whole living animal.

Really, I do. You see, I’m in my first year of veterinary school. Before I got to vet school, I was nervous that I had to memorize everything. I spent a lot of time thinking about how a curriculum could possibly be designed to teach students about the inner workings of…all the mammals…that is, except for humans.

Image courtesy of Natasza A. Kurpios, Cornell University College of Veterinary Medicine

Take the digestive tract, the body’s packing puzzle. It’s essentially a nine meter long tube bounded by the mouth and anus, and it’s the embryo’s job to make that tube fit among all the other structures in the body like the lungs, heart, and liver. You’ve probably seen a picture of the digestive tract at some point. All that stuff in the abdomen looks pretty chaotic.

Here’s a secret that’s not so obvious: the digestive tract is suspended from the back of the body by a thin membrane called a mesentery. Picture a person handling a Chinese yo-yo. Delicate and calculated tugs on the strings can keep the yo-yo straight, or they can make it flip and spin. Similarly in an embryo, the asymmetric growth of cells in the gut tube tug on the mesentery and dictate the way the tube beautifully packs itself into the abdomen.

Proteins always fold up the same way too, but unlike the digestive tract, proteins don’t fold perfectly on the first try. To illustrate what happens, consider social media. You may have noticed that over the last few years, headlines have transitioned from “Campus Reacts to Missouri Protest” to “At First It’s Calm, But Keep Watching and…WHOA.” Extra! Extra! Gone are the days of Newsies beefing up headlines to sell papers on the streets of Manhattan. Now it’s a battle to get the most clicks on the internet, which earned the articles their own buzzword: clickbait.

But how do aggregators on popular clickbait sites write the best headlines? They don’t randomly select words from the dictionary and rearrange them until they sound nice.

The secret to their design, and the driving force of the virality industry, is a proprietary algorithm for “headline testing.” For a given article, over ten different headlines are published (ever notice that the video your friend showed you a week ago was re-posted by another friend under a different title yesterday?). The algorithm determines the headlines that are most popular, and, after reaching a threshold, it picks the best one. After making enough successful headlines, aggregators can spot trends and start predicting success before the articles even get published.

Just like clickbait aggregators, proteins sample a couple of different intermediates before settling into their final state. Unlike clickbait, however, “best” isn’t always the final turnout. Most of the time, the final solution is the one that requires the fewest steps.

The theme is so pervasive throughout the history of life that evolutionary biologists have given it a name: parsimony. This principle drives their ability to describe the relationships between species using phylogenetic trees. They start with a group of organisms and attempt to bin them into related groups based on common aspects of their biology.

For example, monkeys, hippos, and whales all have a four-chambered heart, while lizards and tuna don’t. There are usually a few different ways to sort them, and after sampling the options, they pick the one that’s the simplest. Usually they’re right, until someone unearths a fossil that suggests otherwise.

Even doctors, I’ve now realized, are parsimonious with their diagnoses. The most likely reason why you’re coughing is the one which required your doctor to make the fewest assumptions. Usually they’re right, until a new test suggests otherwise. I suppose that’s why it’s called the practice of medicine; doctors are always encountering new ways to make medical care better.

All you have to do is understand the way that the system interacts with its boundaries, its environment, to see it come together to serve a purpose. That’s exactly what protein biophysicists do to predict molecular conformations, what doctors do to make diagnoses, what clickbait aggregators do to write the best headlines, what evolutionary biologists do to describe phylogenetic relationships, and it’s what I finally did to solve my housemate’s packing puzzle at 2 am five days after opening the box.

It’s a variation of the Slothouber-Graatsma puzzle, and if you’d like to have a hand at it yourself, I wouldn’t suggest clicking that hyperlink.

Before I got to vet school, I was nervous that I had to memorize everything, but they’ve taught us that it’s all just a packing puzzle. Every day I’m learning new types of pattern recognition to put it all together.

Ari, I love all of the connections you make here – who would have imagined that proteins, the GI tract, clickbait, and evolutionary trees would have ended up all together, woven together so seamlessly? Keep up the great work and I can’t wait to read more of your essays.

I just went to a lecture at Harvard where I learned about another packing problem – spheres pack with a density of 64%, meaning that 64% of the space is taken up by the spheres, and the rest is air. It turns out that this observation has many practical applications, in cooking and elsewhere.